Negative Infinity is Smallest Element

Theorem

Let $\left({\overline \R, \le}\right)$ be the extended real numbers with their usual ordering.

Then $-\infty$ is the smallest element of $\overline \R$.

Proof

We have, by definition of the usual ordering on $\overline \R$:

$\forall x \in \overline \R: -\infty \le x$

That is, $-\infty$ is the smallest element of $\overline \R$.

$\blacksquare$

Also see

• Positive Infinity is Greatest Element